82.8.4 problem Ex. 4

Internal problem ID [18752]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Exercises at page 25
Problem number : Ex. 4
Date solved : Tuesday, January 28, 2025 at 12:14:30 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} y^{4}+2 y+\left (x y^{3}+2 y^{4}-4 x \right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 27

dsolve((y(x)^4+2*y(x))+(x*y(x)^3+2*y(x)^4-4*x)*diff(y(x),x)=0,y(x), singsol=all)
 

\[ x -\frac {\left (-y \left (x \right )^{2}+c_{1} \right ) y \left (x \right )^{2}}{y \left (x \right )^{3}+2} = 0 \]

✓ Solution by Mathematica

Time used: 60.238 (sec). Leaf size: 2021

DSolve[(y[x]^4+2*y[x])+(x*y[x]^3+2*y[x]^4-4*x)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}+\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}+\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}-\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}-\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ \end{align*}